A box contains 5 coins: 4 regular coins and 1 fake coin. When a regular coin is tossed, the probability P(head) = 0.5 and for a fake coin, P(head) = 1. You pick a coin at random and toss it twice, and get two heads. The probability that the coin you have chosen is the fake coin is _______. (rounded off to two decimal places)

GATE CSE 2025 Session 1 Q32: Probability the coin is fake after two heads

Engineering Mathematics
1 mark
NAT
No negative marking

A box contains 5 coins: 4 regular coins and 1 fake coin. When a regular coin is tossed, the probability $P(\text{head}) = 0.5$ and for a fake coin, $P(\text{head}) = 1$ . You pick a coin at random and toss it twice, and get two heads. The probability that the coin you have chosen is the fake coin is _______. (rounded off to two decimal places)

Try it: numerical answer: type the value

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Numerical answer. No negative marking.

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Answer: 0.49 to 0.51 (official GATE 2025 final answer key)

Concept used: Bayes' theorem with the law of total probability.

First step: $P(HH) = \tfrac{1}{5} \cdot 1 + \tfrac{4}{5} \cdot \tfrac{1}{4}$ , and $P(\text{fake} \mid HH) = \dfrac{\tfrac{1}{5}}{P(HH)}$ .

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Source: official GATE 2025 Session 1 CS question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

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GATE CSE 2025 Session 1 Q32: Probability the coin is fake after two heads

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